Factor The Polynomial 4 X 4 − 20 X 2 − 3 X 2 + 15 4x^4 - 20x^2 - 3x^2 + 15 4 X 4 − 20 X 2 − 3 X 2 + 15 By Grouping. What Is The Resulting Expression?A. ( 4 X 2 + 3 ) ( X 2 − 5 (4x^2 + 3)(x^2 - 5 ( 4 X 2 + 3 ) ( X 2 − 5 ] B. ( 4 X 2 − 3 ) ( X 2 − 5 (4x^2 - 3)(x^2 - 5 ( 4 X 2 − 3 ) ( X 2 − 5 ] C. ( 4 X 2 − 5 ) ( X 2 + 3 (4x^2 - 5)(x^2 + 3 ( 4 X 2 − 5 ) ( X 2 + 3 ] D. ( 4 X 2 + 5 ) ( X 2 − 3 (4x^2 + 5)(x^2 - 3 ( 4 X 2 + 5 ) ( X 2 − 3 ]

Introduction

Factoring polynomials is an essential skill in algebra, and one of the most effective methods for factoring is by grouping. In this article, we will explore how to factor the polynomial $4x^4 - 20x^2 - 3x^2 + 15$ by grouping, and we will also discuss the importance of factoring in mathematics.

What is Factoring by Grouping?

Factoring by grouping is a method used to factor polynomials that are not easily factored using other methods, such as the difference of squares or the sum and difference of cubes. This method involves grouping the terms of the polynomial in a way that allows us to factor out common factors.

Step 1: Group the Terms

To factor the polynomial $4x^4 - 20x^2 - 3x^2 + 15$, we need to group the terms in a way that allows us to factor out common factors. We can start by grouping the first two terms and the last two terms:

$4x^4 - 20x^2 - 3x^2 + 15 = (4x^4 - 20x^2) - (3x^2 - 15)$

Step 2: Factor Out Common Factors

Now that we have grouped the terms, we can factor out common factors from each group. From the first group, we can factor out $4x^2$:

$(4x^4 - 20x^2) = 4x^2(x^2 - 5)$

From the second group, we can factor out $-3$:

$-(3x^2 - 15) = -3(x^2 - 5)$

Step 3: Combine the Factored Groups

Now that we have factored out common factors from each group, we can combine the factored groups to get the final result:

$4x^2(x^2 - 5) - 3(x^2 - 5) = (4x^2 - 3)(x^2 - 5)$

Conclusion

In this article, we have shown how to factor the polynomial $4x^4 - 20x^2 - 3x^2 + 15$ by grouping. We started by grouping the terms, then factored out common factors from each group, and finally combined the factored groups to get the final result. The resulting expression is $(4x^2 - 3)(x^2 - 5)$.

Importance of Factoring in Mathematics

Factoring is an essential skill in mathematics, and it has many applications in various fields, such as algebra, geometry, and calculus. Factoring allows us to simplify complex expressions, solve equations, and analyze functions. It is also a fundamental concept in many mathematical theories, such as group theory and ring theory.

Common Applications of Factoring

Factoring has many common applications in mathematics, including:

• Solving Equations: Factoring allows us to solve equations by setting each factor equal to zero and solving for the variable.
• Analyzing Functions: Factoring allows us to analyze functions by identifying their roots, maxima, and minima.
• Simplifying Expressions: Factoring allows us to simplify complex expressions by combining like terms.
• Proving Theorems: Factoring is used to prove many mathematical theorems, such as the fundamental theorem of algebra.

Conclusion

Introduction

In our previous article, we explored how to factor the polynomial $4x^4 - 20x^2 - 3x^2 + 15$ by grouping. In this article, we will answer some common questions about factoring by grouping and provide additional examples to help you understand this concept better.

Q&A

Q: What is the difference between factoring by grouping and factoring by difference of squares?

A: Factoring by grouping involves grouping the terms of a polynomial in a way that allows us to factor out common factors, whereas factoring by difference of squares involves factoring a polynomial of the form $a^2 - b^2$.

Q: How do I know which terms to group together when factoring by grouping?

A: When factoring by grouping, you should group the terms in a way that allows you to factor out common factors. This often involves grouping the terms with the same variable or coefficient together.

Q: Can I factor a polynomial by grouping if it has more than two variables?

A: Yes, you can factor a polynomial by grouping even if it has more than two variables. However, you may need to use other factoring methods, such as the distributive property, to simplify the polynomial before factoring by grouping.

Q: How do I know if a polynomial can be factored by grouping?

A: A polynomial can be factored by grouping if it can be written in the form $(a+b)(c+d)$, where $a$, $b$, $c$, and $d$ are polynomials.

Q: Can I factor a polynomial by grouping if it has a negative coefficient?

A: Yes, you can factor a polynomial by grouping even if it has a negative coefficient. However, you may need to use other factoring methods, such as the distributive property, to simplify the polynomial before factoring by grouping.

Q: How do I factor a polynomial with a variable in the denominator?

A: When factoring a polynomial with a variable in the denominator, you should first simplify the polynomial by multiplying the numerator and denominator by the conjugate of the denominator. Then, you can factor the polynomial by grouping.

Q: Can I factor a polynomial by grouping if it has a fraction?

A: Yes, you can factor a polynomial by grouping even if it has a fraction. However, you may need to use other factoring methods, such as the distributive property, to simplify the polynomial before factoring by grouping.

Examples

Example 1: Factoring a Polynomial with Two Variables

Factor the polynomial $x^2y^2 + 2xy^2 + 3y^2$ by grouping.

Solution:

$x^2y^2 + 2xy^2 + 3y^2 = (x^2y^2 + 2xy^2) + 3y^2 = xy^2(x + 2) + 3y^2$

Example 2: Factoring a Polynomial with a Negative Coefficient

Factor the polynomial $-x^2y^2 + 2xy^2 - 3y^2$ by grouping.

Solution:

$-x^2y^2 + 2xy^2 - 3y^2 = (-x^2y^2 + 2xy^2) - 3y^2 = -xy^2(x - 2) - 3y^2$

Example 3: Factoring a Polynomial with a Variable in the Denominator

Factor the polynomial $\frac{x^2y^2}{x^2 + 1} + \frac{2xy^2}{x^2 + 1} + \frac{3y^2}{x^2 + 1}$ by grouping.

Solution:

$\frac{x^2y^2}{x^2 + 1} + \frac{2xy^2}{x^2 + 1} + \frac{3y^2}{x^2 + 1} = \frac{(x^2y^2 + 2xy^2 + 3y^2)}{x^2 + 1} = \frac{y^2(x^2 + 2x + 3)}{x^2 + 1}$

Conclusion

In conclusion, factoring by grouping is a powerful method for factoring polynomials that are not easily factored using other methods. By grouping the terms, factoring out common factors, and combining the factored groups, we can factor complex polynomials and simplify expressions. We hope that this Q&A guide has helped you understand this concept better and provided you with additional examples to help you practice factoring by grouping.